`cot(v - u)` Find the exact value of the trigonometric expression given that sin(u) = -7/25 and cos(v) = -4/5 (Both u and v are in quadrant III.)

Monday, January 19, 2015

$\displaystyle{\cot{{\left({v}-{u}\right)}}}$ Find the exact value of the trigonometric expression given that sin(u) = -7/25 and cos(v) = -4/5 (Both u and v are in quadrant III.)

$\displaystyle{\sin{{\left({u}\right)}}}=-\frac{{7}}{{25}}$

using pythegorean identity,

$\displaystyle{{\sin}}^{{2}}{\left({u}\right)}+{{\cos}}^{{2}}{\left({u}\right)}={1}$

$\displaystyle{{\left(-\frac{{7}}{{25}}\right)}}^{{2}}+{{\cos}}^{{2}}{\left({u}\right)}={1}$

$\displaystyle{{\cos}}^{{2}}{\left({u}\right)}={1}-\frac{{49}}{{625}}=\frac{{{625}-{49}}}{{625}}=\frac{{576}}{{625}}$

$\displaystyle{\cos{{\left({u}\right)}}}=\sqrt{{\frac{{576}}{{625}}}}=\pm\frac{{24}}{{25}}$

Since u is in quadrant III ,

$\displaystyle\therefore{\cos{{\left({u}\right)}}}=-\frac{{24}}{{25}}$

$\displaystyle{{\sin}}^{{2}}{\left({v}\right)}+{{\cos}}^{{2}}{\left({v}\right)}={1}$

$\displaystyle{{\sin}}^{{2}}{\left({v}\right)}+{{\left(-\frac{{4}}{{5}}\right)}}^{{2}}={1}$

$\displaystyle{{\sin}}^{{2}}{\left({v}\right)}+\frac{{16}}{{25}}={1}$

$\displaystyle{{\sin}}^{{2}}{\left({v}\right)}={1}-\frac{{16}}{{25}}=\frac{{{25}-{16}}}{{25}}=\frac{{9}}{{25}}$

$\displaystyle{\sin{{\left({v}\right)}}}=\sqrt{{\frac{{9}}{{25}}}}=\pm\frac{{3}}{{5}}$

since v is in quadrant III,

$\displaystyle\therefore{\sin{{\left({v}\right)}}}=-\frac{{3}}{{5}}$

$\displaystyle{\cot{{\left({v}-{u}\right)}}}=\frac{{\cos{{\left({v}-{u}\right)}}}}{{\sin{{\left({v}-{u}\right)}}}}$

$\displaystyle{\cot{{\left({v}-{u}\right)}}}=\frac{{{\cos{{\left({v}\right)}}}{\cos{{\left({u}\right)}}}+{\sin{{\left({v}\right)}}}{\sin{{\left({u}\right)}}}}}{{{\sin{{\left({v}\right)}}}{\cos{{\left({u}\right)}}}-{\cos{{\left({v}\right)}}}{\sin{{\left({u}\right)}}}}}$

plug in the values of sin(v),sin(u),cos(v) and cos(u),

$\displaystyle{\cot{{\left({v}-{u}\right)}}}=\frac{{{\left(-\frac{{4}}{{5}}\cdot-\frac{{24}}{{25}}+{\left(-\frac{{3}}{{5}}\right)}\cdot-\frac{{7}}{{25}}\right)}}}{{{\left(-\frac{{3}}{{5}}\cdot-\frac{{24}}{{25}}-{\left(-\frac{{4}}{{5}}\right)}\cdot-\frac{{7}}{{25}}\right)}}}$

$\displaystyle{\cot{{\left({v}-{u}\right)}}}=\frac{{\frac{{96}}{{125}}+\frac{{21}}{{125}}}}{{\frac{{72}}{{125}}-\frac{{28}}{{125}}}}$

$\displaystyle{\cot{{\left({v}-{u}\right)}}}=\frac{{\frac{{117}}{{125}}}}{{\frac{{44}}{{125}}}}$

$\displaystyle{\cot{{\left({v}-{u}\right)}}}=\frac{{117}}{{44}}$

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Monday, January 19, 2015

$\displaystyle{\cot{{\left({v}-{u}\right)}}}$ Find the exact value of the trigonometric expression given that sin(u) = -7/25 and cos(v) = -4/5 (Both u and v are in quadrant III.)

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